*Shadows of the Truth: Metamathematics of Elementary Mathematics*, as collaborative. He's asked mathematicians to describe their memories of their childhood understandings and confusions of math, and has woven their stories into a rich tapestry in which he analyzes the learning of mathematics.

I've only read the first chapter so far, and am looking forward to enjoying the rest. I loved his examples of how language affects our mathematical understandings (page 29). The words 'and' and 'or' have particular meanings in mathematics. These meanings can be quite different from the meanings given in a person's natural language:

... what matters in the context of the this book are invisible differences in the logical structure of my students’ mother tongues which may have huge impact on their perception of mathematics. For example, the connective “or” is strictly exclusive in Chinese: “one or another but not both”, while in English “or” is mostly inclusive: “one or another or perhaps both”. Meanwhile, in mathematics “or” is always inclusive and corresponds to the expression “and/or” of bureaucratic slang. In Croatian, there are two connectives “and”: one parallel, to link verbs for actions executed simultaneously, and another consecutive .I also loved his confusion as a child about units (page 17):

When, as a child, I was told by my teacher that I had to be careful with “named” numbers and not to add apples and people, I remember asking her why in that case we can divide apples by people: 10 apples : 5 people = 2 apples. Even worse: when we distribute 10 apples giving 2 apples to a person, we have 10 apples : 2 apples = 5 people.

Where do “people” on the right hand side of the equation come from? Why do “people” appear and not, say, “kids”? There were no “people” on the left hand side of the operation! How do numbers on the left hand side know the name of the number on the right hand side?

This confusion seems to have led him to some of the insights he shares in this book. Enjoy!

I think if we asked elementary and middle school teachers to determine if 10 apples/2 people = 5 apples is correct or not, and to explain why it is right or wrong we would get an insight into their thinking and teaching.... come to think of it, maybe we should take that up to high school algebra teachers as well.

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